In a small bowl, beat 1 egg yolk, 3 tablespoons cream, and ½ teaspoon vanilla extract and set aside. In a food processor, pulse 1¼ cups flour, 1/3 cup sugar, and ¼ teaspoon salt to combine. Scatter the butter over the flour mixture and pulse until the mixture has the texture of coarse cornmeal. With the motor running, add the yolk-cream mixture and process just until the dough comes together, about 45 seconds (do not overprocess). Transfer the dough to a piece of plastic wrap, knead it once or twice, pat it into a neat 5- or 6-inch disk, wrap, and refrigerate until very firm, at least 1 hour and up to 3 days.

Rest the dough at room temperature for about 5 minutes. Roll out the dough between two large pieces of plastic wrap, dusting with flour as needed, into a circle about 14 inches in diameter and ¼-inch thick. Gently transfer the dough to an 11-inch tart pan with a removable bottom. Ease the dough into the corners of the pan, and fold any excess into the sides of the pan, pressing to reinforce the shell. Cover with plastic wrap and place in the freezer for at least 30 minutes.

Adjust the oven rack to the center position and heat the oven to 375 degrees. Place the pan on a baking sheet, line the dough with aluminum foil, and fill with pie weights, enough to reach partway up the sides. Bake for 25 minutes, rotating halfway through baking time. Carefully remove the foil and weights and continue baking until the tart shell is a light golden brown, about 5 minutes longer (cover the edges with foil if they’re browning too much).

Meanwhile, in a bowl, gently toss the sliced apples, 2 tablespoons sugar, and cinnamon. Arrange apples in concentric circles in tart shell and bake until apples just start to soften (the tip of a paring knife poked in will meet with moderate resistance), about 18 minutes, rotating the pan halfway through baking time (cover the edges with foil if they’re browning too much). Remove from the oven and set aside (leave the oven on).

In a medium bowl whisk the remaining egg yolk and 2 tablespoons sugar until very thick and almost white in color, about 2½ minutes. Add the remaining cream, Calvados, remaining vanilla extract, and a pinch of salt and combine well. Sift the remaining flour into the mixture and whisk to combine thoroughly. Strain the custard mixture into the tart over the apples, cover the crust edges with foil, and bake until custard is set and a toothpick inserted into the custard comes out clean, about 30 minutes, rotating the pan halfway through baking time. Sprinkle the tart evenly with the remaining sugar, switch the oven to broil, and with the oven rack about 6 inches below the heating element, broil to brown the top of the tart, about 45 seconds.

Place the tart on a wire rack to cool for at least 40 minutes. Cut it into slices and serve warm or at room temperature.

1 large sweet-tart apple (10 to 12 ounces), cored and cut into matchsticks

¼ cup chopped fresh parsley

½ cup hazelnuts or walnut pieces, lightly toasted and chopped

In a small saucepan over medium-high heat, boil the cider until reduced to 2 tablespoons, about 4 minutes. Scrape the reduced cider into a large bowl, and whisk in the vinegar, mustard, shallot, ¼ teaspoon salt, and pepper to taste. Vigorously whisk in the oil. Taste and adjust the seasoning, if necessary.

Add the beets and apple and toss to coat with the dressing; set aside for flavors to meld, about 10 minutes, then taste and adjust the seasoning, if necessary. Toss in the parsley and nuts. Serve at once.

Sprinkle chicken liberally all over with salt and pepper. Heat oil in a large, straight-sided saute pan over medium-high heat. Place chicken pieces in pan skin side down (do not crowd; brown in batches if necessary) and cook, without moving, until the skin is crisp and golden, about 4 minutes. Turn the chicken pieces and cook, again without moving, about 4 minutes longer. Transfer chicken to a large plate. When cool enough to handle, remove and discard the skin. Discard all but 1 tablespoon of fat from the pan.

Return the pan to the burner, adjust the heat to medium, add the onions and 1 teaspoon salt, and saute until the onions begin to soften and release some liquid, about 6 minutes. Reduce heat to medium-low and continue cooking, stirring and scraping the bottom of the skillet frequently, until the onions are soft, sticky, and golden brown, about 40 minutes longer. Add the garlic, thyme, sage and bay leaves and cook, stirring, until fragrant, about 40 seconds. Add the vinegar and cider, adjust heat to medium-high, bring to a strong simmer, and with a wooden spoon scrape bottom of pan to loosen and dissolve the fond, about 30 seconds. Simmer vigorously, stirring occasionally, until liquid is reduced by about 30 percent, about 3 minutes. Add the chicken with its accumulated juices (reserve the breasts, if using) and return to simmer. Reduce heat to low, cover pan, and simmer gently until the chicken is cooked through and tender, 35 to 45 minutes, turning the chicken over and adding the breasts, if using, after 20 minutes. Remove the chicken to a bowl, tent loosely with foil to keep warm, and set aside. Add about half the parsley to the onions and stir to mix.

Scrape the onions into a strainer set over a bowl and gently fold them to release as much liquid as possible (you should have about 2/3 cup); remove the bay leaves and set the onions aside. Return the liquid to the pan, add the accumulated juices from the resting chicken, bring to a strong simmer over medium-high heat, and simmer for 1 minute. Adjust heat to medium-low, add the butter and melt, stirring constantly to incorporate it into the sauce. Add half the remaining parsley and stir to incorporate. Taste the sauce and adjust the seasoning with salt, if necessary, and pepper to taste.

Spread the onions into a bed on a warmed serving platter, arrange the chicken over the onions, top the chicken with the sauce, sprinkle with the remaining parsley, and serve at once.